Step by step
The question
Question: add 2/(x² − 1) + 1/(x + 1).
1 Factor the bottoms
x² − 1 is a square minus a square.
So it is (x + 1)(x − 1).
We write the bottoms as factors.
2 Find the common bottom
The bottoms are (x + 1)(x − 1) and (x + 1).
The common bottom must have every factor of both bottoms, each one once.
It is (x + 1)(x − 1).
Careful!
We can't add the tops and add the bottoms.
Fractions need the same bottom first.
3 Make the bottoms the same
The first fraction already has it.
The second fraction is missing (x − 1).
We multiply its top and bottom by (x − 1).
(x − 1)/(x − 1) is 1, so the value stays the same.
4 Add the tops
Now one fraction: (2 + (x − 1)) over (x + 1)(x − 1).
5 Open the brackets
A bracket with a plus or nothing in front can just be removed.
Now we have 2 + x − 1.
6 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
2 − 1 is 1.
So the expression is x + 1.
7 Cross out the same
We write the top as factors too: (x + 1).
The same bracket on top and bottom is 1.
Cross them out.
What is left is 1/(x − 1).
The bottoms can't be 0.
So x can't be −1 or 1.
8 Check
Let's check with a number: x = 2.
First in the question.
2/3 plus 1/3 is 1.
Now in our answer.
We get 1 too.
Both are 1.
So 1/(x − 1) is right!
Let's look back.
First: factor the bottoms.
Then: find the common bottom.
Then: make the bottoms the same.
Then: add the tops.
Then: open the brackets.
Then: group like terms.
Then: cross out the same.
Last: check.